Coherent structures in wall turbulence with drag reduction

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POLITECNICO DI MILANO

SCUOLA DI INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Corso di Laurea Magistrale in Ingegneria Aeronautica

Coherent structures in wall turbulence

with drag reduction

Relatore: Prof. Maurizio QUADRIO Correlatore: Dr. Ing. Davide GATTI

Tesi di Laurea di: Emanuele Gallorini Matricola 876564

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Abstract

In this work the physical mechanisms which appear when a flow undergo a drag variation due to a control technique as oscillating wall or streamwise waves of spanwise velocity are studied. To do that 5 Direct Numerical Simulation (DNS) of a turbulent channel flow at Reτ = 200 were used: a first case without forcing,

two cases with oscillating wall and two cases with travelling waves. The first part of the thesis is based on the study of the Coherent Structures (CS) of the flow, in particular of the Quasi-Streamwise Vortices (QSV). The hope is to relate the changes induced by the control on the flow with the variation of the QSV dynamics and subsequently understand how and why the friction is modified. To do that several programs, that allowed us to educe QSV, conditionally average them and extract some relevant physical quantities (as Reynolds stresses), were developed. From that it was possible to confirm the presence of the same Q2 suppression and Q4 enhancement mechanisms underlined in Yakeno et al. [2014] for the vortices under an oscillating wall control, while for the first time their relevance for travel-ling waves was observed. The last part of the work is focused on the proposal of a formula which allows to predict, given the wavenumber kx and the frequency ω

as-sociated to a travelling wave, the friction variation, in order to prove the relevance of the physical phenomena observed during the thesis. To do that we started again from a proposal of Yakeno et al. [2014] which was modified taking into account the GSL acceleration and the QSV-roll phenomenon we observed for the travelling waves, that we related to the modification of the stress state due to ∂w/∂x. An_e alternative formulation, based on ∂2

e

w/∂y2_{, was proposed, observing the striking}

similarity between the drag reduction map of Yakeno et al. [2014] and the repre-sentation of that quantity in the (kx, ω) space as suggested in Duque-Daza et al.

[2012]. Regarding that, we must take into account that this last analysis studies the growth of perturbations in a base flow considering linearized Navier-Stokes equations: this suggest a relevance of linear phenomena for the friction reduction. Key words: DNS, drag reduction, plane channel flow, GSL, Coherent Structures, Quasi-Streamwise Vortices

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Sommario

In questo lavoro vengono studiati i meccanismi fisici che si manifestano quando una corrente subisce una variazione di attrito dovuta ad una tecnica di controllo come la parete oscillante o le onde di velocità trasversale viaggianti in direzione della corrente. Per fare ciò 5 Direct Numerical Simulations (DNS) di canale piano turbolento a Reτ = 200 sono stati utilizzati: un primo caso privo di forzamento,

due casi con parete oscillante e due casi con onde viaggianti. La prima parte della tesi è basata sullo studio delle Strutture Coerenti (CS) presenti nella corrente, in particolare dei Quasi-Streamwise Vortices (QSV). La speranza è quella di mettere in relazione i cambiamenti indotti dal controllo sulla corrente con le variazione della dinamica dei QSV e conseguentemente comprendere come e perché l’attrito viene modificato. Per far ciò sono stati sviluppati diversi programmi che ci hanno con-sentito di estrarre i QSV, farne una media condizionata e ricavare alcune quantità fisiche di rilievo (come gli sforzi di Reynolds). Da ciò è stato possibile confermare la presenza degli stessi meccanismi di soppressione degli eventi Q2 e miglioramento degli eventi Q4 evidenziati in Yakeno et al. [2014] per i vortici sottoposti ad un controllo con parete oscillante, mentre per la prima volta ne è stata osservata la rilevanza per le onde viaggianti. L’ultima parte del lavoro si concentra sulla proposta di una formula che consenta di prevedere, dato il numero d’onda kx e

la frequenza ω associati ad un’onda viaggiante, la variazione di attrito, in modo da provare la rilevanza dei fenomeni fisici osservati nel corso della tesi. Per far ciò siamo partiti nuovamente da una proposta di Yakeno et al. [2014] che è stata modificata tenendo conto dell’accelerazione del GSL e di un fenomeno di rotazione dei QSV osservato per le onde viaggianti che abbiamo legato alla modifica dello stato di sforzo dovuto a ∂w/∂x. Una formulazione alternativa, legata a ∂_e 2w/∂y_e 2 è stata altresì proposta, dopo aver osservato la sorprendente somiglianza tra la mappa di riduzione di attrito di Yakeno et al. [2014] e la rappresentazione di tale quantità nello spazio (kx, ω) come suggerito in Duque-Daza et al. [2012]. A tal

proposito bisogna tener presente che quest’ultima analisi prende in considerazione la crescita di perturbazioni di un flusso base considerate le equazioni di Navier-Stokes linearizzate: ciò suggerisce la rilevanza di fenomeni lineari per la riduzione di attrito.

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Parole chiave: DNS, riduzione di resistenza, corrente in un canale piano, GSL, Strutture Coerenti, Quasi-Stremwise Vortices

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Chapter 1

Intoduction and aim of the work

Friction drag is a major cause of expense in different fields as airplanes, cars, trains and pipelines transports, just to name a few. In order to reduce its economic and ecological impact many works in the last years focused on the introduction of techniques to decrease the friction turbulent drag. Between these, some simple and convenient open loop strategies are the oscillating wall and their generalization in the space, namely streamwise travelling waves of spanwise velocity. Waves designed to optimize the drag reduction have been shown to reduce the friction drag of more than 50% at a small cost, considering that the net energy saving can be more than the 20%. This striking effectiveness brought many researchers to direct their attention on the application of these techniques in different geometries as channel flows (Jung et al. [1992],Quadrio et al. [2009]), pipe flows (Auteri et al. [2010]), boundary layers (Skote [2012]) or even less canonical geometries as cylinders (Zhao et al. [2019]). Most of these works are focused on the research of the most effective actuation parameters, while few others (Ricco et al. [2012],Agostini et al. [2014]), Touber and Leschziner [2012]) examine the details of the fundamental interactions underlying the drag-reduction mechanisms. Anyway, despite from the great effort spent during these years, there is not an agreement on how the control techniques influence the flow and lead to the drag reduction. A promising way in that sense is the one that consider their influence on the Coherent Structures. In Yakeno et al. [2014] the effect of the oscillating wall on the quasi-streamwise vortices for a channel flow geometry is studied and it is found out that it leads to a double mechanism of suppression of Q2 events due to spanwise shear and enhancement of Q4 events due to the pressure strain and vortex tilt that globally produce a drag reduction. The paper ends proposing a formula for the prediction of drag variation based on the considerations developed during the work. The aim of this thesis is to reproduce the results of Yakeno et al. [2014] for the oscillating wall and extend their analysis to the streamwise travelling waves case, underlying the differences on the physical effects introduced by the two techniques and then proposing a possible improved formulation to predict the drag increase or reduction. To do

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that QSV were educed applying the λci scheme proposed in Zhou et al. [1999] and

then conditionally averaged to obtain a vortex representative of the mean behavior of this kind of structures. This work is structured as follow:

• Chapter 2: in this chapter a review of the basic tools needed to understand the work is proposed.

• Chapter 3: in this chapter the adopted λci criterion, as well as the

condition-ally average method, are explained. We also pointed out why we chose this particular criterion and how it simplifies the vortex candidates selection. • Chapter 4: in this chapter we summarize the results of our QSV study,

comparing it with the ones of Yakeno et al. [2014] and presenting the extended analysis in the streamwise travelling waves case.

• Chapter 5: in this chapter we extend the drag reduction prediction formula of Yakeno et al. [2014] to the streamwise travelling waves case and then we analyze the differences between them and the oscillating wall in order to produce an improved attempt to predict the friction variation.

• Chapter 6: in this chapter conclusions and possible future developments are provided.

• Appendices: in these chapters further details and analysis omitted in the main work are shown.

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Chapter 2

Fundamentals and state of the art

This chapter is a review of the fundamental notions necessary to understand the present work. The first section concerns the background of the turbulence and its historical evolution, as done by many authors. In the second section we focus on the turbulent regime of a channel flow, namely the geometry around which this thesis develops. The third section is a glimpse on one of the most relevant topic of this work, that will be extensively discussed in the following chapters: the so-called "Coherent Structures" (CS) and in particular the Quasi-Streamwise Vortices. The fourth and ending section is an overview on the flow control techniques.

2.1 Turbulence: background

Turbulence is said to have a ubiquitus nature: this means that it involves every kind of flow and the interest on this subject is shared by different branch of knowl-edge as physics and engineering of course, but also for example biology, geology and medicine. Despite having a so wide range of application and a lot of researchers studying them, the analytical solution of the equations that describe the fluid mo-tion (and so turbulence), namely Navier-Stokes (NS) equamo-tions, represents an open challenge for mathematicians. From the engineering point of view, the problem of solving NS equations can be faced in two ways: through Direct Numerical Simula-tion or through Reynolds decomposiSimula-tion and Turbulence Models. Both ways have of course positive and negative sides, in particular DNS are computationally very expensive but produces high-fidelity results when the design choices are wisely taken, while the Reynolds decomposition requires the introduction of an empirical model for the Reynolds stresses and so a low reliability of the results, despite being computationally efficient. A pioneering work for the turbulence studies was the one described in Reynolds [1883]. During this experiment Reynolds proved the existence of a laminar and turbulent regime and he discovered that the turbulent properties begin to appear at particular values of a non-dimensional number, called

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nowadays Reynolds number, defined as : Re = U L

ν (2.1)

where U and L are the typical velocity and length scale of the flow, ν is cinematic viscosity of the fluid.

2.2 Channel flow

Figure 2.1: Scheme of the channel flow geometry

The flow studied in this work is a wall bounded flow called plane channel flow (see Fig. 2.1). This is a flow enclosed between two infinite plane walls separated by a distance 2h and driven by a pressure difference in the streamwise direction. Regarding that, there are three possible approach to model the external action necessary to drive the flow: Constant Pressure Gradient (CPG), in which pressure gradient in constant, whereas flow rate fluctuates in time; Constant Flow Rate (CFR), in which flow rate is kept constant, whereas pressure gradient fluctuates in time and finally Constant Power Input (CPI), in which the rate of energy that goes into the system is kept constant. In this work the CPG strategy is used in all the simulations. x, y, z represent respectively streamwise, wall-normal and spanwise directions, while u, v, w the corresponding velocities. At this point some useful quantities can be defined:

Ub = 1 2h Z 2h 0 udy (2.2)

is the bulk velocity. The operator · · · represents the time average. From Ub a

Reynolds number can be defined:

Reb =

Ubh

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Under the hypothesis of incompressibility, the equations that describe the channel flow are the incompressible Navier-Stokes equations:

   ∂u ∂t + (u · ∇)u + ∇P − ν∇ 2_{u = 0} ∇ · u = 0 (2.4) The Reynolds decomposition of the velocity can be introduced:

u(x, t) = u(x) + u0(x, t). (2.5) Averaging Eq. 2.4 with relation 2.5 we obtain the Reynolds Averaged Navier-Stokes equations (RANS). From RANS equations, exploiting the proprieties of the channel flow, a relation valid for the shear stress is obtained:

τ = ρν∂u ∂y − ρu

0_v0 _(2.6)

where −ρu0_v0 _{are the so called Reynolds stresses. From the non-slip boundary}

condition velocity is 0 at the wall, so the shear stress at the wall, namely τw:

τw = ρν ∂u ∂y w . (2.7)

In the near-wall region viscous effect dominate over inertia, we can then define an inner scaling using the wall shear stress τw and the cinematic viscosity ν:

uτ =

p

τw/ρ δν = ν/uτ tν = δν/uτ = ν/u2τ (2.8)

Analyzing a channel flow, it is common to scale the variables with the quantities of Eq.2.8. The resultant adimensionalized values are called viscous units. In this work, from now on ∗ will represent dimensional quantities, while viscous units scaled quantities will be represented without any symbol:

u = u ∗ uτ y = y ∗ δν (2.9) A distinction in different regions, based on the non dimensional distance from the wall, can be done:

• y < 5: viscous sub-layer: the effect of the viscosity overwhelm the one of inertia.

• 30 ≤ y ≤ 50 logarithmic region: the mean streamwise velocity profile follows a logarithmic law.

• 5 ≤ y ≤ 30 buffer layer: links the logarithmic region and viscous sub-layer. • y > 50 outer layer: there is no direct effect of the viscosity in the shear

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2.3 Coherent Structures

Figure 2.2: Scheme of quadrant interactions

As seen in the previous sections turbulence is dominated by chaos. A large amount of research effort has been spent, starting from the 1960s, on the attempt to find a deterministic frame in the chaotic background, the so called coherent structures. These structures are difficult to be described: a possible definition attempt is the one of Robinson [1991]:

A coherent motion is a three dimensional region of the flow over which at least one fundamental flow variable exhibits significant corre-lation with itself or with another variable over a range of space and/or time that is significantly larger than the smallest local scales of the flow. From the practical point of view coherent structures can be identified through flow visualization, by conditional sampling technique or through eduction techniques as the ones described in Chapter 3. Following Pope [2000] the different motivations to analyze turbulent structure are:

• to look for order in the turbulent chaos

• try to explain patterns seen in flow visualization

• try to explain important flow mechanisms in term of elemental structures • to identify relevant structures with a view to modifying them in order to

achieve engineering goals such as reduction of drag and augmentation of heat transfer.

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The last two points are in particular the ones faced in this work. The coherent structure approach to turbulent flows should of course not be viewed as a con-ceptual or academic interest but it is of profound practical significance for the understanding, design and safety of natural and man-made systems involving tur-bulence. Anyway, despite from the fact that this kind of method has been useful to understand some topics about turbulence and have injected new momentum and excitement into research, it is also true that, as stated in Hussain [1986], it raised many more questions than it has answered. That happened because there are many structures within the random background, and their three dimensional de-terministic and stochastic interactions are far from clear and unlikely to be simple. Between the different kinds of coherent structures we can recognize:

• Low-Speed Streaks (LSS) in the y < 10 region • Ejections of near wall low-speed fluid

• Sweeps of high-speed fluid towards the wall

• Various vortices (the most important of which are Quasi-Streamwise Vortices) • Inclined shear layers.

A fundamental instrument for the CS study is the so called quadrant analysis introduced by Wallace et al. [1972] (see Fig. 2.2). The Reynolds stresses can be divided in 4 quadrant contribution depending on the sign of u0 and v0. Whenever the instantaneous product u0_v0 _{overcomes a threshold, an event is said to occur.}

We refer to a specific event with the letter Q followed by a number (1, 2, 3, 4) indicating the quadrant. Q2 events are characterized by low streamwise speed (u0 < 0) and by a movement away from the wall (v0 > 0), so they represents ejections, while Q4 events describe high-speed fluids (u0 > 0), moving downward (v0 < 0) called sweeps. Q1 (u0 > 0, v0 > 0) and Q3 (u0 < 0, v0 < 0) events are called interactions mode. In wall-bounded turbulence sweeps and ejections are prevalent, leading to positive values of u0_v0 _{(Kim et al. [1987]), responsible for the increase}

of the skin-friction drag in turbulent flows (see Eq.4.1) and for the production of turbulent kinetic energy (u0_v0_{du/dy) through the interaction with the mean shear.}

As previously said coherent structures interact one with the other. An exam-ple of interaction nowdays understood, at least qualitatively, is the so called wall cycle. It is given by the mutual influence of LSS and QSV: the LSS are generated via lift-up of low-speed fluid near the wall by the induced v of (mature) stream-wise vortices, (Waleffe [1995]), while QSV are generated by streak instability, in particular through the streak transient growth mechanism described in Schoppa and Hussain [2002]: this kind of cycle is said to be self-sustaining QSV generates LSS and vice-versa, and moderate perturbations (w_rms0 = 0.5) are enough to lead to the generation of new vortices.

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2.3.1 The QSV

Figure 2.3: The QSV educed in Jeong et al. [1997]

According to Jeong and Hussain [1995] the concept of vortices is old as hydro-dinamics, yet, an accepted definition of vortex is still lacking. A possible practical definition is the one of Robinson [1991]:

A vortex exist when instantaneous streamlines mapped onto a plane normal to the vortex core exhibit a roughly circular or spiral pattern, when viewed from a reference frame moving with the center of the vortex core.

Vortex dynamics, which govern the evolution and interaction of CS and coupling of CS with background turbulence, is promising not only for understanding turbu-lence phenomena such as entrainment and mixing, heat and mass transfer, chem-ical reaction and combustion, drag, and aerodynamic noise generation, but also for viable modeling of turbulence. There are different reasons to study the vortex dynamic in a turbulent flow, between the others:

• In a boundary layer, a vortex with an orientation different from the wall-normal, can transport mass and momentum through the velocity gradient and can have an important role in turbulent phenomena as Reynolds stresses production (Bernard et al. [1993])

• Vortices are a particularly important class of CS, since they are among the most coherent of turbulent motions and tend to be persistent in the absence of destructive instabilities

• Strong vortices work as a source for pressure disturbances by virtue of their low-pressure cores and the high pressure regions they can induce in the nearby flow.

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The idea of quasi-streamwise vortex has a particular relevance in the CS back-ground, as confirmed in Jeong et al. [1997]. In this work a numerically simulated channel flow is studied and CS near the wall (y ≤ 60) are educed using λ2

cri-terion. After a conditional average (see Sec. 3.4) it is shown that the dominant structures in the analyzed domain are elongated quasi-streamwise vortices inclined 9◦ in the vertical (x, y) plane and tilted ±4◦ in the horizontal (x, z) plane. The vortices of alternating sign overlap in x as a staggered array and moreover there is no indication near the wall of the popular hairpin vortices(see Acarlar and Smith [1987],Zhou et al. [1999]), not only in the educed data but also in instantaneous fields. The tilting of the angle in the (x, z) planes is particularly important in generating negative pressure strain p∂u/∂x that distributes fluctuations of kinetic energy from streamwise direction to spanwise and wall-normal sustaining in that way the CS. The same tilting is also responsible for kinked low-speed streaks and internal shear layers with negative ∂u/∂x, both commonly observed but not pre-viously linked to near-wall CS.

2.4 Flow control

Figure 2.4: Flow control strategies

Flow control is probably the hottest topic in fluid mechanics due to its techno-logical and economic implications. It contains different fields as transition, sepa-ration and turbulence control with a variety of objective that can be for example reducing drag, increasing lift, increasing mixing or reducing noise. This particular work is focused in turbulence control for drag reduction. There are different flow control strategies as can be seen in Fig. 2.4, usually divided in passive and active.

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Passive techniques are systems that don’t need energy to work, as riblets (Luchini [1993]), compliant coatings, transegrity fabric and injection of polymers, between the others. On the other side, active techniques require an energy input and they can be divided in predetermined, meaning that control parameters are decided a priori, and reactive, where control depends on the state of the flow and its action is adjusted continuously based on measurements of some kind. Reactive control can be feedback or feed-forward: in the first case the measured variable is the same to be controlled while in the second one measure and control differ. The oscillating wall and streamwise travelling waves of spanwise velocity strategies analyzed in this work belong to active predetermined control strategies.

2.4.1 Oscillating wall

This technique was first introduced by Jung et al. [1992]. The authors started from the observation (Bradshaw and Pontikos [1985], Moin et al. [1990]) that when a two dimensional turbulent boundary layer is subjected to a sudden spanwise pressure gradient the production of turbulent quantities, including Reynolds shear stresses and turbulent kinetic energy, is suppressed and these changes has been attributed to modifications of coherent structures due to the new transverse strain. If the imposed constant gradient is constant, the suppression of turbulence is only temporary, with the flow returning to a new state with different orientation and higher Reynolds number, so authors proposed the possibility of a sustained control of turbulence by spanwise oscillations either of the spanwise cross-flow or the channel walls. In this work the second case in taken into consideration, in particular from the practical point of view it consist in imposing a spanwise velocity at the wall:

ww(x, 0, z, t) = ww(x, 2Reτ, z, t) = A cos(ωt) (2.10)

where A is the amplitude of the wall oscillation and ω = 2π/T is its frequency, with T period of the oscillation. It can be noted that the oscillating wall can be considered a particular case of streamwise travelling wave of spanwise velocity with kx = 0.

2.4.2 Streamwise travelling waves of spanwise velocity

This technique was introduced in Quadrio et al. [2009] combining the oscillating wall of Jung et al. [1992] and the stationary streamwise-modulated spanwise oscil-lations studied in Viotti et al. [2009] (see Fig.2.5). A spanwise velocity is imposed at the wall:

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Figure 2.5: Sketch of a streamwise travelling wave of spanwise velocity

where A is the amplitude of the wall oscillation, kx is the streamwise wavenumber

and ω is the frequency. The wave moves forward or backward in the streamwise direction with a phase speed c:

c = ω kx

(2.12) One of the critical points of this work is trying to clarify the mechanism that produce drag reduction, that is believed to be caused by the thin transversal boundary layer generated by the wall waves, both unsteady and wall-modulated, the so called Generalized Stokes Layer (GSL). An analytical solution for the GSL can be found for a laminar flow, while in presence of turbulence the laminar GSL solution describes well the space-averaged spanwise flow if the phase speed of the waves is sufficiently different from the turbulent convection velocity and that the time scale of the forcing is smaller than the life time of near-wall structures. If these conditions are verified, drag reduction is found to scale well with GSL thickness, so that the laminar solution can be used for the turbulent flow analysis.

2.4.3 Phase decomposition of the velocity field

Velocity can be decomposed in different contribution in order to analyze the flow field, in particular, as proposed in Yakeno et al. [2014], the phase-average velocity can be considered: huii(y, φ) = 1 N N −1 X n=0 1 Lx 1 Lz Z Lx 0 Z Lz 0 ui(x, y, z, φ + 2πn)dxdz, (2.13)

with φ = ωt in case of oscillating-wall and φ = kxx − ωt in travelling waves

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The instantaneous velocity field can than be decomposed in:

ui(y, φ) = huii(y, φ) + u00i(x, y, z, t), (2.14)

with u00_i(x, y, z, t) random velocity fluctuation. The phase averaged quatities can be further decompose into a time-averaged and a periodic component:

huii(y, φ) = ¯ui(y) +uei(y, φ). (2.15) The instantaneous flow field can than be rewritten as:

ui(x, y, z, t) = ¯ui(y) +uei(y, φ) + u

00

i(x, y, z, t). (2.16)

In this work 8 phases are taken into consideration, for both oscillating-wall and streamwise travelling waves cases, with φ = {0π₄ π₄ 2π₄ 3π₄ 4π₄ 5π₄ 6π₄ 7π₄ }. From the practical point of view huii can be obtained for the oscillating wall case

av-eraging together velocity fields corresponding to a time t = t + nT , after being averaged along x and z direction. Moreover, being the flow field symmetric on the y direction, the upper and the lower part of the channel can be considered together to obtain a more precise mean velocity profile. For what concerns the travelling waves every flow field considered contains 4 waves, so that all the phases are present, but they out of sync. It is possible to perform a circular shift of the fields in the x direction due to its periodic nature and bring all of them in phase. At that point an average in performed considering all the four waves contained in all the field and the upper and lower part of the channel, due to its symmetry on the y direction.

2.4.4 Drag reduction measure

The drag reduction can be quantified through different parameters, as the friction coefficient Cf = 2τw ρU2 b . (2.17)

Changes of Cf are quantified in terms of Cf,0, the friction coefficient of the

reference turbulent flow using a parameter R: R = Cf,0− Cf

Cf,0

= 1 − Cf Cf,0

, (2.18)

that can be related to the variation of bulk velocity ∆Ub = Ub − Ub,0 through

the relation: ∆Ub = s 2 Cf − s 2 Cf,0 = s 2 Cf,0 1 √ 1 −R − 1 . (2.19)

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A different parameter to characterize the drag reduction is the vertical shift ∆B of the mean velocity profile. According to the classical theory Pope [2000] the mean streamwise velocity profile u(y) presents a region, between the buffer layer and the viscous sub-layer, where it follows a logarithmic law:

u = 1

kln y + B (2.20)

where k is the von Kármán constant and B the additive constant or near-wall intercept. In the region far from the wall the velocity-defect law is valid. It describes the difference between the local mean velocity and the centerline mean velocity Uc: Uc− u = − 1 k ln y∗ h + B1, (2.21)

where B1 is a flow dependent constant representing the difference between the

ac-tual centerline velocity Ucand the velocity obtained by extrapolating the

velocity-defect law up to the centerline. If Eq. 2.20 and Eq.2.21 are summed together and Ucis substituted with Ub taking advantage of the relation Uc= Ub+ 1/k and using

the definition 2.17 for Cf, the following relation can be obtained:

s 2 Cf = 1 k ln Reτ + B + B1− 1 k (2.22)

The change of ∆B has traditionally been used as an indicator of the roughness-induced drag change (see e.g. Clauser [1956]; Jiménez [2004]) and of the drag reduction due to the presence of riblets (see e.g. Luchini et al. [1991]; Garcia-Mayoral and Jiménez [2011]). In Gatti and Quadrio [2016] is shown that a control based in streamwise travelling waves modify the mean velocity profile through ∆B. At that point the friction law 2.22 can be used to obtain a dimensionless relation between Reτ, ∆B and R: s 2 Cf − s 2 Cf,0 = 1 kln Reτ Reτ,0 + ∆B (2.23)

In the present work the DNS are performed under CPG, so the Reτ = Reτ,0, and

the Eq.2.23 is equal to Eq. 2.19 and so: ∆B = ∆Ub = s 2 Cf,0 1 √ 1 −R − 1 (2.24) Than ∆B and ∆Ub can be indifferently used to estimate the drag reduction, as

well asR once relation 2.24 is used. Once these concepts are fixed, we can proceed with the drag reduction estimation, that in the present work is done by ∆B (or ∆Ub) computation. The main advantage in using ∆B instead of R in that the

first is a Re independent quantity, while the second, being a function of Cf and

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Chapter 3

QSV eduction

In order to identify a QSV, different techniques have been experimented during the years such as the Q criterion (Hunt et al. [1988]), λ2 criterion (Jeong and

Hussain [1995]), ∆ criterion (Chong et al. [1990]) and swirling strength (or λci)

criterion (Zhou et al. [1999]). All of these methods exploit the local analysis of the velocity gradient tensor ∇u: the velocity field around a point in r position can be expressed as a Taylor expansion trucated at the first order:

u(r + δr) = u(r) + ∇uδr + O(||δr||2). (3.1) The ∇u characteristic equation is:

λ3+ P λ2+ Qλ + R = 0 (3.2)

where P , Q and R the invariants of the velocity gradient tensor. All of the previously cited methods take advantage of 3.2 and its properties and the interested reader can consult Chakraborty et al. [2005] for a compact review. In this work in particular the λci criterion is used for some peculiarities that will be clarified in

paragraph 3.3.

3.1 Swirling strength criterion: λ

ci

The swirling strength criterion uses the imaginary part of the complex conju-gate eigenvalues of ∇u to identify vortices. For an incompressible flow, the first invariant P = −∇ · u is 0 and the discriminant for 3.2 can be expressed as:

∆ = (1 2R) 2 + (1 3Q) 3 . (3.3)

∇u has a real eigenvalue and a pair of conjugated complex eigenvalues when ∆ is positive. In that case ∇u can be decomposed in :

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∇u =νr νcr νci   λr λcr λci −λci λcr  νr νcr νci −1 ,

where λr and νr are the real eigenvalue and the real eigenvector, λcr, λci, νcr

and νci are respectively the real and imaginary parts of the conjugated complex

eigenvalues with the corresponding eigenvectors. In a local curvilinear coordinate system (y1, y2, y3) defined by {νr, νcr, νci} translating with the fluid particle, the

local streamlines are expressed as:

y(1) = Crexp λrt, (3.4)

y(2) = exp λcrt[Cc(1)cos(λcit) + Cc(2)sin(λcit)], (3.5)

y(3) = exp λcrt[Cc(2)cos(λcit) − Cc(1)sin(λcit)], (3.6)

with C1, C (1)

c and Cc(2) constants. In Eq. 3.4, 3.5 and 3.6 the fact that the

local flow is stretched or compressed along νr while it swirls in the νcr− νci plane

is shown. This pattern for the flow streamlines suits very well with the working definition of vortex in Robinson [1991] quoted in subsection 2.3.1.

3.2 Comparison between λ

ci

and λ

2

At that point the problem of defining a λci threshold over which a vortex can

be identified is still open. In the literature the swirling strength method is not widely used and so a commonly accepted value for the threshold still does not exist. The same is not valid for the λ2 criterion, where a value of λ2 ≤ −0.002

is usually adopted to identify a vortex core. Starting from this consideration, a possible way to relate λci and λ2 is proposed in Chakraborty et al. [2005]. In

particular, considering:

λci ≥ (λci)th = (3.7)

(λcr/λci) ≤ (λcr/λci)th= δ (3.8)

The relation:

λ2 ≤ (λ2)th= −2, (3.9)

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Figure 3.1: Ratio between the volume of detected vortices considering or not the con-straint −(λcr/λci)th ≤ λcr/λci ≤ (λcr/λci)th. This is a section of inverse spiralling

compactness domain, considering that its maximum is 30.16

3.2.1 Influence of inverse spiralling compacteness

The condition of Eq. 3.8 is not directly a constraint on λcibut on the ratio λcr/λci,

also called inverse spiralling compactness: this parameter measures the spatial extent of the local spiralling motion and it has been introduced to enhance the swirling strength criterion. In order to determine if inverse spiralling compactness is relevant or not in our work, we analyzed its influence on the choice of the threshold for the swirling strength: we identified in the flow field, for different values of λci, the λci ≥ (λci)th regions in the presence or not of the constraint

−(λcr/λci)th ≤ λcr/λci ≤ (λcr/λci)th. Then we computed the number of points

that respect the inequalities in a field for the two cases and in the end we divided the value found applying both the constraints for the one with λci ≥ (λci)th alone.

As can be seen in Fig. 3.1 the inverse spiralling compactness parameter seems to not be relevant in our case apart from a very thin region close to 0, that could anyway be an image artifact depending on the discretization of the parameters. For this reason, and for the fact that there is not a general agreement on the possible value of the threshold (λcr/λci)th we decided to not consider it.

3.2.2 Swirling strength tuning

We performed a tuning process to determine a threshold for the swirling strength producing results as much as possible similar to the well established λ2 criterion.

Considering (λ2)th= −2 = −0.002 (see for example Jeong et al. [1997]) we chose

the value of (λci)th =

√

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Figure 3.2: Percentage of points equally identified (as vortices or not) with λ2 and

swirling strength criteria with respect to the value of (λci)th for 5 flow fields. Optima

are represented by circles.

suggested in Eq. 3.9. First of all we defined the parameter EP as the percentage

of points of a flow field identified in the same way (vortex or not) in the case of λ2 and swirling strength criteria: calculating EP means to compute λci and λ2 in

all the field, verify where the conditions λci ≥ (λci)th and λ2 ≤ (λ2)th are satisfied,

find the number of points where the two criteria give the same result and compute their percentage with respect to the total number of points. Fig. 3.2 shows how EP changes with (λci)th in 5 different random fields: as can be seen the optimum

values are similar, and a value of (λci)th = 0.145 is chosen.

3.3 Vortex identification

In Fig. 3.4 and 3.5 the regions of the channel with λci≥ (λci)th for a random flow

field are represented. Qualitatively, the viscous sub-layer contains no vortices, that are concentrated in the buffer region and while their distribution over wall-normal direction will be discussed more deeply in Sec. 4.3, this first result, in agreement with other works as Jeong et al. [1997], gave us a confirmation about the accuracy of our work. After that, the λci field was transformed in a binary

image (value 1 if λci ≥ (λci)th, otherwise 0) and the regions with adjacent pixels,

namely connected regions, were isolated and identified as vortex candidates. For each vortex candidate different properties were computed:

• vortex center: there exist different possibilities for the definition of the vortex center, in this work was defined as the point, within a connected region, where λci assumes the maximum value;

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Figure 3.3: Example of positive ω00_x vortex with negative sense of rotation

• vortex orientation: this issue is the first one where the swirling strength criterion shows an advantage compared to the others: in fact λci method

provides a directional information given by the space orientation of the real eigenvector νr. As stated in Sec. 3.1, this is the direction along which the

vortex is stretched or compressed and we considered it, evaluated in the vortex center previously defined, as the vortex axis. In that way the spatial orientation of a vortex candidate can be easily computed: in particular we considered θxy (inclination angle), θzx (tilting angle) and θyz, that represent

the angles formed between νr and x direction in the (x, y) plane, z direction

in the (x, z) plane and y direction in the (y, z) plane respectively;

• vortex length: we used the information provided by νr to compute the length

of a vortex. Usually this parameter is computed through an advancing method that, starting from a relevant point, as the vortex center, follows a certain property, as could be the maximum value of λci, in subsequent

(y, z) planes. This kind of evaluation is reliable but could be quite expensive from the computational point of view. In order to make our program as effi-cient as possible, we chose a linearized method, where the word linear means in that case that the vortex was approximated with a line. In particular con-sidering the smallest (x, y, z) direction box containing the vortex candidate, we defined its length as the distance between the two points formed by the intersection of the box with the straight line, passing from the vortex center, having the direction of νr evaluated again in the vortex center. The

lin-earization could be an issue for very curved vortex, as hairpin vortices, that anyway seems to be a negligible part of the whole number of the detected structures (see Jeong et al. [1997]),

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0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Z Axis 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 X Axis 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Z Axis 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 X Axis 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Z Axis 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 X Axis 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Z Axis 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 X Axis

Figure 3.4: λ_ci> (λci)th regions: bottom view

0 100 200 300 400 Y Axis 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 X Axis 0 200 400 Y Axis 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 X Axis 0 100 200 300 400 Y Axis 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 X Axis 0 200 400 Y Axis 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 X Axis

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• vortex type: in order to consider the different behavior of vortices with dif-ferent sense of rotation in response to the introduction of a flow control tech-nique, we divided them in four different groups: negative or positive rotating vortices at the lower wall, negative or positive rotating vortices at the upper wall. We defined the sense of rotation as the sign of the scalar product of the random vorticity fluctuation vector ω00 and νr, after all the real eigenvectors

are aligned in order to point towards positive x direction. This approach was chosen because various research (Gao et al. [2011], Bernard et al. [1993], Zhou et al. [1999]) noted that the local vorticity vector is not always aligned with the direction of the structures in turbulent wall bounded flows. In order to understand the meaning of this, considering the extreme example of Fig. 3.3, can be seen that ω_x00 > 0, but, since ω00· νr < 0, the vortex sense of

rotation is negative and so the commonly accepted distinction based on the sign of ω00_x could lead to wrong conclusions. Having said that, it is also true that in Jeong et al. [1997], for example, the mean inclination angle of a vortex in the x − y plane is found to be 9◦, and so the differences between various distinction methods should not be particularly relevant.

To avoid the presence of noise and small QSV not completely developed, a threshold in the length of a candidate, in order to assume it as a vortex, was introduced. In particular, we considered a limit of 50 wall unit in the reference case described in section 4.1. This corresponds to a threshold of ca. 0.866 in the cumulative distribution function, meaning that we take into account just the 13.4% of the vortex candidates. For oscillating wall and streamwise travelling waves of spanwise velocity cases, we exploited the same threshold in the cumulative distribution function rather than the simple length one because the vortex length could be affected by the introduction of a flow control technique and in that way we compared the same set of vortices, independently from variation of properties and number.

3.4 Conditional sampling

Once the quasi-streamwise vortices were educed from the flow field we proceeded with the conditional sampling operation. Our scheme is inspired to the ones pro-posed by Jeong et al. [1997] and Yakeno et al. [2014], but presents some particular features in order to be applied to the oscillating-wall and streamwise travelling waves cases. First of all, a neighborhood (a point higher and one lower) of the maximum of the wall-normal QSV distribution is selected and just vortices belong-ing to this set are averaged together: this is done to avoid smearbelong-ing the resultbelong-ing QSV by vortices located at a lower or higher position. The selection of the sam-pling wall-normal height has to be repeated every time a new case is taken into consideration since it is not possible to determine a priori if the control action

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will affect the y vortex position. After that, the flow fields are shifted along the periodic x and z directions to arrange the vortices center in the same point of the domain, in our case corresponding to the channel middle point xm and zm:

from the practical point of view, given the vortex center positions xv and zv, this

mean to shift the field of the quantities ∆x = xm − xv and ∆z = zm − zv. Once

the vortices centers occupy the same position, the velocity and pressure fluctu-ations fields (u00 and p00) are averaged together. Note that multiple vortices are associated to the same flow field: it is than necessary to sample them singularly, considering independently over the average procedure the flow field a number of times corresponding to the vortices associated to it. During the average operation is necessary to note some details, depending on the case taken into consideration: • Reference case: according to Jeong et al. [1997] QSV with different sense of rotation are tilted in a symmetric way around (x, y) plane (see Fig. 2.3). So, taking the positive sense of rotation as a reference, negative rotating vortices has to be flipped around z direction to make to two coincide. Moreover, in the sampling process we take into consideration vortices associated to both the lower and upper walls being one upside down with respect to the other. We therefore flip upper wall vortices around y direction but this is not enough: upper wall vortices with positive sense of rotation would have, once flipped, the same tilting orientation of lower wall vortices with negative sense of ro-tation, so that is necessary a second flip around z direction. The correctness of these geometric consideration can be confirmed from the mathematical point of view: according to the definition of subsection 2.3.1 the streamlines generated by a vortex should exhibit a roughly circular or spiral pattern if mapped onto a plane normal to the vortex core. Considering that velocity fields induced by vortices with a different sense of rotation would counteract during the averaging process producing a different result with respect to the one expected by a QSV, it is necessary to sample structures with the same sense of rotation, namely same sign of ω_x00. Taking for example into consid-eration positive rotating upper wall vortices: flipping around y and z axis mean to apply a transformation for which yo = −y and vo = −v and than

another for which zo = −z and zo = −z, where the subscript o indicate the

terms before the transformation. Since: ω00_x,o= ∂w 00 o ∂yo − ∂v 00 o ∂zo , (3.10)

after the geometrical transformation: ωx = ∂w00 ∂y − ∂v00 ∂z = −∂w00 o −∂yo −−∂v 00 o −∂zo = ω00_x,o. (3.11) This means that, as expected, the sign of ω00_x has not changed during the process. With similar reasoning con be shown that the sign of ω_x00 changes for the transformations associated to negative rotating vortices.

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• Controlled case: adopting a flow control technique as oscillating-wall or streamwise travelling waves means to apply time or space-time dependent strategies. In particular, we recall that these are periodic methods character-ized by a phase φ, with φ = 2πt/T in the first case and φ = kxx − ωt in the

second. A control with a certain phase φ affect in different ways the different structures: vortices with a positive tilting angle, namely lower-wall positively rotating and upper-wall negatively rotating ones, withstand a forcing velocity field equal to the one affecting vortices with negative tilting angle, namely lower-wall negatively rotating and upper-wall positively rotating ones, at a phase φ + π since the two sets are symmetric with respect to the (x, y) plane. During the sampling operations is then necessary to consider positively tilted structures at the phase φ with the negatively tilted ones at a phase φ + π and vice versa. Besides that, it is also necessary to flip the vortices as described for the uncontrolled reference case.

Once the flow fields have been average together, the results will be a velocity field uc_(∆x

c, y, ∆zc) and a pressure field pc(∆xc, y, ∆zc), with the superscript c

meaning a conditional average quantity, ∆xc = x − xv and ∆zc= z − zv indicating

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Chapter 4

Modification of QSV in a

drag-reduced channel flow

The goal of this chapter is to describe the results of our database analysis and to compare it with the existing literature, proving its correctness. The cases taken into consideration are:

• Reference case: Channel flow without any kind of flow control

• Oscillating walls: Channel flow with the introduction of a flow control technique based on the oscillating wall (Jung et al. [1992]). We took into consideration 2 different cases present in Yakeno et al. [2014], both with A = 7: T = 75 and T = 250.

• Streamwise traveling waves of spanwise velocity: Channel flow with the introduction of a flow control technique based on streamwise traveling waves of spanwise velocity (Quadrio et al. [2009]). We took into consideration 2 different cases: one with drag reduction (A = 7, ω = 0.0238, kx = 0.01 )

and the other with drag increase (A = 7, ω = 0.12, kx = 0.01 ).

4.1 Database building

The present work is based on the post-processing of five sets of DNS data: one is a fully developed channel flow without any control strategy applied and the re-maining four with a drag reduction or increase technique embedded. The database

Case Nf ields Reτ Reb Lx/h Lz/h Nx× Ny× Nz ∆t

REF 90 200 3180 4π 2π 256 × 193 × 256 0.1178 Table 4.1: Parameters for the simulations of the reference case

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Case Nf ields Reτ Reb Lx/h Lz/h Nx× Ny× Nz A T ∆t

OW1 376 200 3674 4π 2π 256 × 193 × 256 7 75 0.0938

OW2 368 200 3414 4π 2π 256 × 193 × 256 7 250 0.0781 Table 4.2: Parameters for the simulations of the oscillating wall case

Case Nf ields Reτ Reb Lx/h Lz/h Nx× Ny× Nz A ω kx ∆t

TW1 100 200 3982 4π 2π 256 × 193 × 256 7 0.0238 0.01 0.0974 TW2 100 200 2930 4π 2π 256 × 193 × 256 7 0.12 0.01 0.1102 Table 4.3: Parameters for the simulations of the streamwise travelling waves of spanwise velocity case

were produced thanks to the DNS code written by Luchini and Quadrio (Luchini and Quadrio [2006]) in which Navier-Stokes equation are projected in the v − η space, with v wall-normal component of the velocity and η wall-normal compo-nent of the vorticity as proposed in Kim et al. [1987], and solved using a pseudo-spectral method. The simulations were run under CPG strategy with an imposed Reτ = 200, with Reτ = uτh/ν (h = 1, uτ = 1 and ν = 0.005). After the

implemen-tation of the control system the flow rate undergo a variation, increasing in case of drag reduction or decreasing in case of drag increase: as a consequence the Reb

based on Ub changes. The size of the computational domain was 4π ×2h×2π along

streamwise, wall-normal and spanwise direction. Nx = Ny = 256 Fourier modes

discretized the periodic x and z directions while along y direction an hyperbolic tangent distribution with N y = 193 nodes was chosen in order to obtain a more defined grid close to the wall. Staring from the initial condition, the flow fields underwent a transient phase in which CF L = 1 was imposed to determine the time step ∆t. After the transient ∆t was fixed in the oscillating wall case in order to sample the phases considered during this work (φ = {0π₄ π₄ 2π₄ 3π₄ 4π₄ 5π₄ 6π₄ 7π₄ }). The time step, averaged in case of CF L = 1 or imposed in oscillating wall case, as well as the number of fields saved Nf ields for each simulation are shown in tables

4.1, 4.2 and 4.3. Pressure was computed solving the Poisson equation. The λci,

λcr, λ2 and ωx00 fields, necessary to educe the quasi-streamwise vortices, were

com-puted starting from the DNS databases using a CPL code written for this work exploiting the concepts expressed in chapter 3.

4.2 Quadrant contribution of Reynolds shear stresses

Fukagata, Iwamoto and Kasagi derived in Fukagata et al. [2002] a mathematical relationship between different dynamical contributions and Cf under CFR

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OW1 OW2 TW1 TW2 −2 −1 0 1 2 3 4 ∆ RQ i Q1 Q2 Q3 Q4 ∆Qi

Figure 4.1: Quadrant contribution to Ub calculated base to the FIK identity at A+= 7.

For each quantity the uncontrolled value is subtracted. The sum of the four contributions ∆Qi is the total variation of Ub from the uncontrolled case.

Ub = Reτ 3 − Z Reτ 0 1 − y ∗ Reτ (−u0_v0_)dy _(4.1)

The objective of the control is to reduce the second term: in fact Reτ/3 is

iden-tical to the flow rate for a laminar flow, while the second one represent turbulent contribution. Being the integral positive in uncontrolled turbulent flows, it is ob-vious that effect of turbulence is to reduce the flow rate from the laminar value, as expected. Another point to be underlined is that the turbulence contribution is a weighted integral of −u0_v0_{, with a weighting function that linearly decays from the}

wall to the channel center: this confirms that the Reynolds shear stresses closer to the wall have a deeper effect on Ub compared to the farther ones. For what

concerns the Reynolds stresses of Eq. 4.1, they can be rewritten since:

u0 =_eu + u00, (4.2)

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u0_v0 _{= g}_u0_v0 ₌₍ _^ e u + u00₎₍ e v + v00₎ = fu_e_ev + gu00 e v + gv00 e u + gu00_v00 =u_e_ev + u00_v00 = u00_v00_, (4.3)

where the relations gu00

e

v = gv00

e

u = _ev = 0, valid due to the geometry of the flow and to the average properties, are exploited. Following Yakeno et al. [2014] Eq. 4.1 can be written considering the contribution to the Reynolds stresses of the 4 quadrant separated: Ub = Reτ 3 + 4 X i=1 R_QiDN S, (4.4)

where RDN S_Qi represents the contribution from i th quadrant of the Reynolds shear stress u0_v0 RDN S_Qi = − Z Reτ 0 1 − y ∗ Reτ (−u00_v00₎ Qidy (4.5)

The superscript DNS indicates that the data are directly taken from flow fields, without considering the conditional average. Since the contributions of Q2 and Q4 events (u00_v00₎

Q2,4 < 0, they are the responsible of the reduction of Ub from the

laminar to the turbulent regime. In Fig. 4.1 ∆RDN S

Qi = RDN SQi − (RDN SQi )REF, with

subscript REF indicating the reference case, is represented for the different cases

taken into consideration in this work: OW1, OW2, TW1 and TW2. Since the first term of Eq. 4.1 is the same with and without flow control:

∆Ub = Ub− (Ub)REF = 4

X

i=1

∆RDN S_Qi . (4.6)

We can observe that the most relevant contribution to ∆Ubis due to Q2 and Q4

events. For what concerns the oscillating wall cases, the result obtained are similar to the ones of Yakeno et al. [2014]: ∆RDN S₂ is always positive, meaning that Q2 events are suppressed, with their contribution to ∆Ub that appears to be the most

relevant in the cases taken into consideration. Regarding sweep events, they seem to have a lower contribution to drag reduction, but they can be either suppressed or enhanced depending on the period of the oscillation T . For this reason the optimum oscillation period is determined by the trade-off of the suppression of Q2 events and the enhancement of Q4 events that appears at high T . Our analysis is also extended to the streamwise traveling waves of spanwise velocity: in case of

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Figure 4.2: Educed vortex in the reference case: side view (upper figure) and bottom view (bottom figure).

drag reduction (TW1), the contributions are similar, although higher, to the ones of the oscillating wall case at the optimal oscillation frequency (OW1), meaning that sweep and ejections are both suppressed, with an higher contribution to drag reduction given by Q2 events. In case of drag increase (TW2) we can observe that both sweep and ejection are enhanced, leading to a decrease of Ub.

4.3 QSV eduction results

The results of the educed QSV in the reference case can be compared with tho ones of Jeong et al. [1997]. We recall the the authors used the λ2 criterion, imposing

a first threshold on the streamwise direction on the range 10 ≤ y ≤ 40 in order to catch fully developed CS and a second one in the tilting angle of ±30◦. As can be seen in Tab. 4.4, despite the differences between the two adopted criteria, the results are strikingly similar. The only parameter that remarkably differs in

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Case Tilting angle Inclination angle ycenter Vortex length

REF 2.55◦ 9.67◦ 22.37 99

Jeong et al. [1997] 4◦ 9◦ 20 200

Table 4.4: Parameters for the simulations of the oscillating wall case

Case REF OW1 OW2 TW1 TW2

Position 22.37 22.37 18.99 23.57 14.90 Table 4.5: Position of the maxima in the y-distribution

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the two cases is the vortex average length. This inconsistency can be explained by two factors: firstly the method used to compute the quantity is different, since we exploited the linearized method explained in Sec. 3.3 while Jeong, Hussain, Schoppa and Kim used an iterative method based on the local maxima of −λ2 in

subsequent (y, z) planes and secondly we imposed a length threshold of 50 wall units for the QSV candidates in the reference case, while in the other work is 150. In Fig. 4.2 the sketch with the educed vortex is shown. In Fig. 4.3 the wall normal distribution of the vortices for the different control cases, after having excluded the shortest as reported in 3.4, is plotted. This parameter is particularly important since for every case is necessary to find the maximum of the distribution in order to select the height of the QSV that are conditionally averaged. As disclosed in Sec. 3.4, differently from Yakeno et al. [2014], the maxima do not coincide in the different cases. Moreover, in the TW2, hence the drag increase case, the shape of the distribution is different compared to the drag reduction cases, being more flattened toward low y values. Another observation can be done looking at Tab. 4.5: the position of the maxima decreases with the decreasing of the effectiveness of the control technique and this qualitatively agrees with the fact that the thickness of the GSL introduced by the control technique is related to the drag reduction (see Quadrio and Ricco [2011] and Cimarelli et al. [2013]). Furthermore is quite straightforward looking at 4.1 that QSV at different heights will have a different weight on the drag reduction: the term 1 − y/Reτ which multiply the Reynolds

stresses is linearly dependent on y, this means that higher CS contributes less to ∆RQ2 and ∆RQ4 (and to the decrease of Ub) with respect to the ones placed in a

lower position and this agrees again quite well with the observation on the maxima of the vortices wall normal distribution.

4.4 u

00

v

00

analysis on wall-normal direction

In Fig. 4.4 and 4.5 the Reynolds stresses −u00_v00 _{over y are represented. Some}

important features can be extracted from these figures. First of all the changes on the curves with respect to the reference case are qualitatively located in the y < 5 and y > 30 regions and this is expected since Eq. 2.6 in valid for a channel flow: in the viscous sublayer u = y, while in the logarithmic region the log-law of the wall is valid and as already seen the introduction of the control affect the u profile with a ∆B shift, so that in these two regions ∂u/∂y is not modified and as a consequence (or maybe not a consequence but a cause since QSV are mostly located in 10 < y < 30 region) u00_v00 _{is forced to remain equal to its uncontrolled}

value. In the cases where drag reduction is obtained the total Reynolds stresses lie in lower level compared to the reference case (as expected) apart from a small region (y < 10) of the OW2; the profiles are more flattened and shifted forward in wall normal direction the more the control is effective. This can be related with the fact that in TW1 and OW1 cases the vortices are moved upward as their

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Figure 4.4: −u00_v00 _{contribution for the oscillating cases: red lines represents Q1 events,}

blue lines Q2, green lines Q3, magenta lines Q4 and black lines the total Reynolds stresses. Continuous lines represent the analyzed case, while dotted lines represent REF.

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Figure 4.5: −u00_v00 _{contribution for the traveling waves cases: red lines represents Q1}

events, blue lines Q2, green lines Q3, magenta lines Q4 and black lines the total Reynolds stresses. Continuous lines represent the analyzed case, while dotted lines represent REF.

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Event OW1 OW2 TW1 TW2 T = 75 T = 250 Q2 suppression 8/8π 4/8π 6/8π 6/8π 6/8π 4/8π Q4 enhancement 14/8π 6/8π 10/8π 12/8π 10/8π 6/8π

Table 4.6: Phases in which Q2 suppression and Q4 enhancements are most noticeable. OW1, OW2, TW1 and TW2 are the cases of analyzed in this work, T = 75 and T = 250 are the ones of Yakeno et al. [2014].

contribution to the Reynolds stresses, but the same is not valid for the OW2 case, where the QSV are moved downward: the vortices displacement on wall-normal direction is not then the only mechanism affecting the Reynolds stresses, since the introduction of a control technique influence the QSV dynamics as described in Yakeno et al. [2014]. Suppression of Q2 and enhancement of Q4 are present and they can be easily seen from the figures: in OW1 and TW1 cases both Q2 and Q4 are suppressed, with the first more relevant than the second, in OW2 case Q2 is suppressed but this positive effect in counteracted by a moderate enhancement of Q4 while in the TW2 case we can only appreciate an increase of −u00_v00 _{due to}

both Q2 and Q4. Modification of quadrant contribution to Reynolds stresses can be related to the vortex dynamics and it will be done in the next section.

4.5 Conditional average results

Fig. 4.6, 4.7, 4.8 and 4.9 show the (vc, wc_{) field for 8 different phases at the position}

xc= 0 (which corresponds to the vortex center) for OW1, OW2, TW1 and TW2

cases, together with thew profile. In addition to the global considerations done in_e the previous part of the chapter, it can be clearly seen that the presence of the flow control modify in a periodic way the distribution and the intensity of Q2 and Q4 events. The conditionally-averaged Reynolds shear stress fields obtained through our analysis in OW1 and OW2 can be compared with the ones shown in Yakeno et al. [2014] looking at table 4.6. The Q2 suppression and Q4 enhancement occur at the same phase in the OW2–T = 250 analysis, while in OW1–T = 75 some differences exist. In the same table the data for TW1 and TW2 are collected. Comparing the images, Q2 and Q4 events detected in this work appear to be more intense with respect to the ones of the paper: this could be the result of the fact that the used QSV eduction scheme is different (λci and not λ2) and that we took

advantage of a length threshold during the vortices selection, while in Yakeno et al. [2014] no constraint on the length is used. A last observation can be made looking how Fig. 4.9 differs with respect to the others: in particular in Fig. 4.6, 4.7 and 4.8 the shape of the Q2 and Q4 events contours clearly change, but the wall normal position of their maxima remains qualitatively the same in all the phases, while in the TW2 case their height clearly change, like if the vortex were rolling around x

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0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.21.4 1.6 1.82 2.22.4 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.2 1.4 1.6 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.4 1.6 1.822.22.4 0.2 0.2 0.4 0.4 0.6 0.6 0.81 1.2 1.4 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.81 1 1.2 1.41.6 1.8 2 2.2 0.2 0.2 0.4 0.4 0.6 0.8 1 1.2 1.4 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.2 1.4 1.6 0.2 0.2 0.4 0.4 0.6 0.6 0.81 1.21.4 0.2 0.2 0.4 0.4 0.6 0.6 0.8 11.21.4 0.2 0.2 0.2 0.4 0.40.6 0.6 0.8 0.8 1 1.2 1.41.6 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1.2 1.41.6 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.4 1.61.8 2 2.2 2.4 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1.2 1.4 1.6 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.41.6 1.8 22.22.4 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1.2 1.4 1.61.8

Figure 4.6: The conditionally averaged flow field with oscillation control; velocity vectors (vc, wc) and weighted Q2 and Q4 contributions in the y − z plane of ∆x_c= 0 at different phases for T+_{= 75. Dashed and solid lines represents Q2 and Q4 events.}

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0.2 0.2 0.4 0.4 0.60.8 1 1.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.2 0.4 0.4 0.6 0.8 1 1.21.4 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.4 1.6 1.8 0.2 0.2 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.4 1.6 1.8 0.2 0.2 0.4 0.4 0.6 0.60.8 0.8 1 1 1.2 1.4 1.61.82 2.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.4 1.61.8 0.2 0.2 0.4 0.4 0.6 0.60.8 0.8 1 1 1.21.41.6 1.822.22.4 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.21.41.6 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.21.41.6 1.82 2.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.21.41.61.8 0.2 0.2 0.4 0.40.6 0.6 0.8 1 1.21.4 1.6 1.8 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.21.41.61.8 0.2 0.2 0.4 0.4 0.6 0.811.2 1.41.6 0.2 0.2 0.4 0.4 0.6 0.6 0.8 11.21.41.61.8

Figure 4.7: The conditionally averaged flow field with oscillation control; velocity vectors (vc, wc) and weighted Q2 and Q4 contributions in the y − z plane of ∆xc= 0 at different

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0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 0.2 0.2 0.4 0.4 0.6 0.6 0.81 1.21.41.6 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.21.41.61.8 22.2 0.2 0.2 0.4 0.4 0.60.81 1.2 0.2 0.2 0.4 0.4 0.6 0.8 1 1.21.4 0.2 0.2 0.4 0.60.81 1.2 0.2 0.2 0.4 0.6 0.8 1 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.2 1.41.6 0.2 0.20.4 0.4 0.6 0.81 1.2 0.2 0.2 0.4 0.4 0.6 0.60.8 0.8 1 1 1.2 1.4 1.6 1.8 0.2 0.20.4 0.4 0.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.4 1.6 1.8 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.4 1.6 1.8 22.2 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.21.4 1.61.8 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.21.4 1.61.822.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.21.41.6 1.8

Figure 4.8: The conditionally averaged flow field with streamwise traveling waves control; velocity vectors (vc, wc) and weighted Q2 and Q4 contributions in the y − z plane of ∆xc = 0 at different phases for TW1. Dashed and solid lines represents Q2 and Q4

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0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.21.4 1.6 1.8 22.2 0.2 0.2 0.4 0.4 0.6 0.8 1 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.21.41.6 1.8 2 2.2 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 0.2 0.2 0.4 0.4 0.6 0.6 0.8 1 1.21.4 1.61.8 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 11.2 1.4 0.2 0.2 0.4 0.60.8 0.2 0.2 0.2 0.4 0.4 0.60.81 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1.21.4 1.61.8 0.2 0.2 0.4 0.4 0.60.8 11.21.41.61.8 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 1.4 1.4 1.6 1.8 2 2.2 0.2 0.2 0.4 0.4 0.6 0.81 1.21.4 1.61.8 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 1.4 1.4 1.61.8 2 2.2 0.2 0.2 0.4 0.4 0.60.81 1.21.41.61.8 0.2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 1.41.6 1.822.2 0.2 0.2 0.40.60.81

Figure 4.9: The conditionally averaged flow field with streamwise traveling waves control; velocity vectors (vc, wc) and weighted Q2 and Q4 contributions in the y − z plane of ∆xc = 0 at different phases for TW2. Dashed and solid lines represents Q2 and Q4